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Transfinite induction
Transfinite induction is an extension of that is applicable to well-founded classes such as an ordinal notation, an ordinal, the class \(\textrm{On}\) of ordinals, and the class \(V\) of sets. It can be used in proving the statements about ordinals such as comparisons of fast-growing functions, and in constructing maps on \(\textrm{On}\) or \(V\) such as elementary operators of ordinals and the rank of sets. Definition Let \(P(\alpha)\) be a predicate on an element \(\alpha\) of a well-founded class \((X,<)\). Suppose that for all \(\alpha \in X\), if \(P(\beta)\) holds for all \(\beta \in X\) satisfying \(\beta < \alpha\), then \(P(\alpha)\) also holds. Then transfinite induction tells us that \(P\) is true for all \(\alpha \in X\). In the case \((X,<) = (\textrm{On},\in)\), although transfinite induction does not refer to whether \(\alpha\) is a successor ordinal or not, a proof is frequently devided in to the following three cases: *Zero case: prove that \(P(0)\) holds. *Successor case: prove that \(\forall \alpha \in \textrm{On} : P(\alpha) \rightarrow P(\alpha+1)\) (for any successor \(\alpha\), \(P(\alpha)\) implies \(P(\alpha+1)\)) holds. *Limit case: prove that \(\forall \lambda \in \textrm{On} \setminus \{0\} : (\forall \beta \in \lambda : \lambda \neq \beta+1) \rightarrow ((\forall \beta \in \lambda : P(\beta)) \rightarrow P(\lambda))\) (for any non-zero limit ordinal \(\lambda\), \(P(\beta)\) implies \(P(\lambda)\)) holds. As axiom schema In ZFC set theory, transfinite induction is a valid deduction. In other words, for any predicate \(P(\alpha)\) on an element \(\alpha\) of a well-founded class \((X,<)\), the closed formula "for all \(\alpha \in X\), if \(P(\beta)\) holds for all \(\beta \in X\) satisfying \(\beta < \alpha\), then \(P(\alpha)\) holds" is provable in \(\textrm{ZFC}\) set theory. This statement, which includes the quantification of the formula \(P\) and the class \((X,<)\), itself is not a theorem in \(\textrm{ZFC}\) set theory, but is a metatheorem, i.e. a theorem in the metatheory, on the provability of the transfinite induction schema. In ZFC set theory, the axiom of regularity implies \(V = \textrm{WF}\), where \(\textrm{WF}\) denotes von Neumann universe, and hence the well-foundedness of \((V,\in)\). Therefore transfinite induction ensure the metatheorem that for any predicate \(P(x)\) on a set \(x\), the closed formula "for all set \(x\), if \(P(y)\) holds for all \(y \in x\), then \(P(x)\) holds" is provable in \(\textrm{ZFC}\) set theory. On the other hand, when we deal with a weaker set theory such as KP set theory, the metatheorem does not necessarily hold. Therefore the transfinite induction on \((V,\in)\) or its restriction to a specific class of predicates can be non-trivial axiom schema. When we deal with an arithmetic such as Peano arithmetic, then the well-foundedness of the formalisation of a recursive well-orderring in the metatheory is not necessarily provable in the arithmetic. Therefore the transfinite induction on a recursive well-orderring does not necessarily hold for the arithmetic. Therefore the transfinite induction on a recursive well-orderring can be non-trivial axiom schema. Indeed, when the metatheory is a set theory, we can define the notion of proof-theoretic ordinal using the provability of axiom schema for (premitive) recursive well-orderrings whose ordinal type is bounded by a given ordinal. Note that the precise definition of proof-theoretic ordinal is quite complicated, and is not literally given as the supremum of ordinals \(\alpha\) in the metatheory such that the transfinite induction on the formalisation of a recursive well-orderring of ordinal type bounded by \(\alpha\) holds for the arithmetic. Examples There are many functions and hierarchies defined by transfinite induction: * The functions in ordinal arithmetic, e.g. the addition, the multiplication, the exponentiation, and so on * Veblen function and its multivariable and transfinite extensions * The enumeration functions of classes of ordinals, e.g. the enumeration \(\alpha \mapsto \Omega_{\alpha}\) of cardinals, the enumeration \(\alpha \mapsto I_{\alpha}\) of weakly inaccessible cardinals and their limits, the enumeration \(\alpha \mapsto M_{\alpha}\) of weakly Mahlo cardinals and their limits, and so on * Fast-growing hierarchy * Hardy hierarchy * Slow-growing hierarchy * von Neumann hierarchy * constructible hierachy See also Category:Concepts